Pseudo-hermitian Random Matrix Theory: Theory & Practice

Pseudo-hermitian random matrices (PHRM) form a new class of matrix models lying between the classical Wigner-Dyson ensembles of hermitian matrices and the non-hermitian Ginibre ensembles. As is the case with the more familiar hermitian random matrix ensembles, PHRMs also have many potential physical applications. These applications range from spectral statistics of the Dirac operator in QCD, through that of linearized Liouvillians around multidimensional saddle points, all the way to statistical description of complex optical systems with balanced gain and loss.
Pseudo-hermitian matrices are hermitian with respect to an indefinite metric over some vector space. Consequently, their eigenvalues are either real or come in complex conjugate pairs.
Ensembles of pseudo-hermitian random matrices could be thought of as probability measures over generators of the non-compact classical Lie algebras, in much the same way classical hermitian random matrices ensembles comprise probability measures over the classical compact algebras.
In this talk I will explain the physical motivation for pseudo-hermitian random matrix theory and present explicit numerical and analytical results pertaining to the average eigenvalue spectrum of a concrete pseudo-hermitian random matrix model in the large-N limit.